Out of curiosity, I'm looking for some examples of Lie groups that are torsion-free.
For some reason (and perhaps there is a good reason), most torsion-free groups I've heard of are discrete. One Lie example that comes to mind is $\mathbb{R}^n$. Also, if $G$ is a Lie group and its torsion subgroup, call it $\Gamma$, is both normal and discrete, then $G/\Gamma$ provides another source of examples.
Question 1: What are some more examples of torsion-free Lie groups?
Question 2: Are there any examples that have non-trivial compact Lie subgroups?
Best Answer
The group of real upper triangular matrices with $>0$ diagonal entries is a good example of a group without elemnt of finite order.
If a Lie group contains a compact group , this subgroup is a compact Lie group, as every closed subgroup of a Lie group is a Lie group (Chevalley). But a compact Lie group is either finite (hence torsion) or contains a maximal torus, hence $T^1$, the group of complex number of modulus 1, and therefore contains elemnets of order $n$ for every $n$.